Mathematical Physics Vol 1

3.2 Vector analysis

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3.2 Vector analysis

Problem24 Prove that two conjugated systems of vectors represent two noncoplanar systems of the same orientation.

Problem25 If A and B are differentiable functions of scalar u , prove that a) d d u ( A · B )= A · d B d u + d A d u · B , b) d d u ( A × B )= A × d B d u + d A d u × B .

Proof a)

( A + ∆ A ) · ( B + ∆ B ) − A · B ∆ u

d d u

( A · B )= lim ∆ u → 0

=

∆ u → 0

∆ u

A · ∆ B + ∆ A · B + ∆ A · ∆ B ∆ u

∆ B ∆ u

∆ A ∆ u ·

∆ A · ∆ B

A ·

B +

lim ∆ u → 0

= lim

+

=

d B d u

d A d u ·

= A ·

B ,

+

∆ A · ∆ B ) / ∆ u = 0.

where we used lim ∆ u

→ 0 (

b)

( A + ∆ A ) × ( B + ∆ B ) − A × B ∆ u =

d d u

( A × B )= lim ∆ u → 0

∆ u → 0

∆ u

A × ∆ B + ∆ A × B + ∆ A × ∆ B ∆ u

∆ B ∆ u

∆ A ∆ u ×

∆ A × ∆ B

A ×

B +

lim ∆ u → 0

= lim

+

=

d B d u

d A d u ×

= A ×

B ,

+

∆ A × ∆ B ) / ∆ u = 0.

where we used lim ∆ u

→ 0 (

Problem26 If a vector function A has a constant magnitude ( | A | is independent of the variable t , and the components A x , A y and A z are functions of t ) prove that A and d A d t are orthogonal if d A d t ̸ = 0.